What is the expected value of a random variable following a lognormal distribution derived from a normal distribution with mean μ and standard deviation σ?

The mathematical approach

Even when there is no drift, so that μ = 0, the expected value of the lognormal distribution is affected by the volatility.
Thus, to simulate paths with truly zero drift, we have to subtract .5*σ² from the drift.
So now we have learned something many people overlook or forget: when you are simulating lognormal returns, you always have to subtract away half the variance from the drift. This is far and away the most common mistake in simulating returns.

The empirical approach

Just take those 1,000 paths we simulated and compute the average

Also see: Lessons From The .50 Delta Option (Moontower)